Page:Annales de mathématiques pures et appliquées, 1826-1827, Tome 17.djvu/127

(168)$\left\{{\begin{aligned}&\int _{0}^{\varpi }{\frac {1-r\operatorname {Cos} .p}{1-2r\operatorname {Cos} .p+r^{2}}}.{\frac {\operatorname {Cos} .{\frac {a}{2}}(\varpi -p)}{\left(\operatorname {Sin} .{\frac {p}{2}}\right)^{a}}}.\operatorname {d} p=2^{a-1}.\varpi \left[1+(1-r)^{-a}\right],\\\\&\int _{0}^{\varpi }{\frac {r\operatorname {Sin} .p}{1-2r\operatorname {Cos} .p+r^{2}}}.{\frac {\operatorname {Sin} .{\frac {a}{2}}(\varpi -p)}{\left(\operatorname {Sin} .{\frac {p}{2}}\right)^{a}}}.\operatorname {d} p=2^{a-1}.\varpi \left[1-(1-r)^{-a}\right].\end{aligned}}\right.$

(169)$\int _{0}^{\varpi }{\frac {1-r\operatorname {Cos} .p}{1-2r\operatorname {Cos} .p+r^{2}}}.\operatorname {l} .\operatorname {Tang} .{\frac {p}{2}}\operatorname {d} p={\frac {\pi }{2}}\operatorname {l} \left({\frac {1-r}{1+r}}\right).$

(170)$\left\{{\begin{aligned}&\int _{0}^{\varpi }{\frac {1-r\operatorname {Cos} .p}{1-2r\operatorname {Cos} .p+r^{2}}}.\operatorname {l} \operatorname {Cos} .{\frac {p}{2}}\operatorname {d} p={\frac {\pi }{2}}\operatorname {l} \left({\frac {1+r}{4}}\right),\\\\&\int _{0}^{\varpi }{\frac {r\operatorname {Sin} .p}{1-2r\operatorname {Cos} .p+r^{2}}}p\operatorname {d} p=\varpi \operatorname {l} (1+r).\end{aligned}}\right.$

(171)$\int _{0}^{\varpi }{\frac {1-r\operatorname {Cos} .p}{1-2r\operatorname {Cos} .p+r^{2}}}.\operatorname {l} .\operatorname {Sin} .{\frac {p}{2}}\operatorname {d} p={\frac {\pi }{2}}\operatorname {l} \left({\frac {1-r}{4}}\right).$

(172)$\left\{{\begin{aligned}&\int _{0}^{\varpi }{\frac {1-r\operatorname {Cos} .p}{1-2r\operatorname {Cos} .p+r^{2}}}.{\frac {\operatorname {l} .\operatorname {Cos} .{\frac {1}{2}}p}{\left({\frac {p}{2}}\right)^{2}+\left(\operatorname {l} .\operatorname {Cos} .{\frac {p}{2}}\right)^{2}}}\operatorname {d} p\\&\qquad \qquad \qquad \qquad ={\frac {\pi }{1-r}}-{\frac {\pi }{2}}\left[{\frac {1}{\operatorname {l} (2)}}+{\frac {1}{\operatorname {l} (2)-\operatorname {l} (1+r)}}\right],\\\\&\int _{0}^{\varpi }{\frac {r\operatorname {Sin} .p}{1-2r\operatorname {Cos} .p+r^{2}}}.{\frac {{\frac {1}{2}}p}{\left({\frac {p}{2}}\right)^{2}+\left(\operatorname {l} .\operatorname {Cos} .{\frac {p}{2}}\right)^{2}}}.\operatorname {d} p\\&\qquad \qquad \qquad \qquad =-{\frac {\pi }{2}}\left[{\frac {1}{\operatorname {l} (2)}}-{\frac {1}{\operatorname {l} (2)-\operatorname {l} (1+r)}}\right]\,;\end{aligned}}\right.$

et ainsi du reste.

On trouvera, au contraire, pour $r^{2}>1,$

(173)$\left\{{\begin{aligned}&\int _{0}^{\varpi }{\frac {1-r\operatorname {Cos} .p}{1-2r\operatorname {Cos} .p+r^{2}}}.\left(\operatorname {Tang} .{\frac {p}{2}}\right)^{a}\operatorname {d} p={\frac {\pi }{2\operatorname {Cos} .{\frac {a\pi }{2}}}}\left[1-\left({\frac {r-1}{r+1}}\right)^{a}\right],\\\\&\int _{0}^{\varpi }{\frac {r\operatorname {Sin} .p}{1-2r\operatorname {Cos} .p+r^{2}}}.\left(\operatorname {Tang} .{\frac {p}{2}}\right)^{a}\operatorname {d} p={\frac {\pi }{2\operatorname {Sin} .{\frac {a\pi }{2}}}}\left[1-\left({\frac {r-1}{r+1}}\right)^{a}\right].\end{aligned}}\right.$